Download Chapter 3, Part A

Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University
© 2002 South-Western /Thomson Learning
Slide 1
Chapter 3
Descriptive Statistics: Numerical Methods






Measures of Location
Measures of Variability
Measures of Relative Location and Detecting
Outliers
Exploratory Data Analysis
Measures of Association Between Two Variables
The Weighted Mean and
Working with Grouped Data
x
Slide 2
Measures of Location





Mean
Median
Mode
Percentiles
Quartiles
Slide 3
Example: Apartment Rents
Given below is a sample of monthly rent values ($)
for one-bedroom apartments. The data is a sample of 70
apartments in a particular city. The data are presented
in ascending order.
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 4
Mean


The mean of a data set is the average of all the data
values.
xIf the data are from a sample, the mean is denoted by
.
 xi
x
n

If the data are from a population, the mean is
denoted by m (mu).
 xi

N
Slide 5
Example: Apartment Rents

Mean
 xi 34, 356
x

 490.80
n
70
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 6
Median


The median is the measure of location most often
reported for annual income and property value data.
A few extremely large incomes or property values
can inflate the mean.
Slide 7
Median



The median of a data set is the value in the middle
when the data items are arranged in ascending order.
For an odd number of observations, the median is the
middle value.
For an even number of observations, the median is
the average of the two middle values.
Slide 8
Example: Apartment Rents

Median
Median = 50th percentile
i = (p/100)n = (50/100)70 = 35.5
Averaging the 35th and 36th data values:
Median = (475 + 475)/2 = 475
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 9
Mode




The mode of a data set is the value that occurs with
greatest frequency.
The greatest frequency can occur at two or more
different values.
If the data have exactly two modes, the data are
bimodal.
If the data have more than two modes, the data are
multimodal.
Slide 10
Example: Apartment Rents

Mode
450 occurred most frequently (7 times)
Mode = 450
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 11
Percentiles


A percentile provides information about how the
data are spread over the interval from the smallest
value to the largest value.
Admission test scores for colleges and universities
are frequently reported in terms of percentiles.
Slide 12
Percentiles

The pth percentile of a data set is a value such that at
least p percent of the items take on this value or less
and at least (100 - p) percent of the items take on this
value or more.
• Arrange the data in ascending order.
• Compute index i, the position of the pth percentile.
i = (p/100)n
• If i is not an integer, round up.
The pth percentile is
the value in the ith position.
• If i is an integer, the pth percentile is the average of
the values in positions i and i+1.
Slide 13
Example: Apartment Rents

90th Percentile
i = (p/100)n = (90/100)70 = 63
Averaging the 63rd and 64th data values:
90th Percentile = (580 + 590)/2 = 585
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 14
Quartiles




Quartiles are specific percentiles
First Quartile = 25th Percentile
Second Quartile = 50th Percentile = Median
Third Quartile = 75th Percentile
Slide 15
Example: Apartment Rents

Third Quartile
Third quartile = 75th percentile
i = (p/100)n = (75/100)70 = 52.5 = 53
Third quartile = 525
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 16
Measures of Variability


It is often desirable to consider measures of
variability (dispersion), as well as measures of
location.
For example, in choosing supplier A or supplier B we
might consider not only the average delivery time for
each, but also the variability in delivery time for each.
Slide 17
Measures of Variability





Range
Interquartile Range
Variance
Standard Deviation
Coefficient of Variation
Slide 18
Range



The range of a data set is the difference between the
largest and smallest data values.
It is the simplest measure of variability.
It is very sensitive to the smallest and largest data
values.
Slide 19
Example: Apartment Rents

Range
Range = largest value - smallest value
Range = 615 - 425 = 190
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 20
Interquartile Range



The interquartile range of a data set is the difference
between the third quartile and the first quartile.
It is the range for the middle 50% of the data.
It overcomes the sensitivity to extreme data values.
Slide 21
Example: Apartment Rents

Interquartile Range
3rd Quartile (Q3) = 525
1st Quartile (Q1) = 445
Interquartile Range = Q3 - Q1 = 525 - 445 = 80
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 22
Variance


The variance is a measure of variability that utilizes
all the data.
It is based on the difference between the value of
each observation (xi) and the mean (x for a sample, 
for a population).
Slide 23
Variance


The variance is the average of the squared differences
between each data value and the mean.
If the data set is a sample, the variance is denoted by
s2.
s2 

2
(
x

x
)
 i
n 1
If the data set is a population, the variance is denoted
by  2.
 ( xi   ) 2
 
N
2
Slide 24
Standard Deviation




The standard deviation of a data set is the positive
square root of the variance.
It is measured in the same units as the data, making
it more easily comparable, than the variance, to the
mean.
If the data set is a sample, the standard deviation is
s  s2
denoted s.
If the data set is a population, the standard deviation
is denoted  (sigma).
  2
Slide 25
Coefficient of Variation


The coefficient of variation indicates how large the
standard deviation is in relation to the mean.
If the data set is a sample, the coefficient of variation
is computed as follows:
s
(100)
x

If the data set is a population, the coefficient of
variation is computed as follows:

(100)

Slide 26
Example: Apartment Rents

Variance

s 
n 1
2

( xi  x ) 2
 2 , 996.16
Standard Deviation
s  s2  2996. 47  54. 74

Coefficient of Variation
s
54. 74
 100 
 100  11.15
x
490.80
Slide 27
Measures of Relative Location
and Detecting Outliers




z-Scores
Chebyshev’s Theorem
Empirical Rule
Detecting Outliers
Slide 28
z-Scores


The z-score is often called the standardized value.
It denotes the number of standard deviations a data
value xi is from the mean.
xi  x
zi 
s



A data value less than the sample mean will have a zscore less than zero.
A data value greater than the sample mean will have
a z-score greater than zero.
A data value equal to the sample mean will have a zscore of zero.
Slide 29
Example: Apartment Rents

z-Score of Smallest Value (425)
xi  x 425  490.80
z

 1. 20
s
54. 74
Standardized Values for Apartment Rents
-1.20
-0.93
-0.75
-0.47
-0.20
0.35
1.54
-1.11
-0.93
-0.75
-0.38
-0.11
0.44
1.54
-1.11
-0.93
-0.75
-0.38
-0.01
0.62
1.63
-1.02
-0.84
-0.75
-0.34
-0.01
0.62
1.81
-1.02
-0.84
-0.75
-0.29
-0.01
0.62
1.99
-1.02
-0.84
-0.56
-0.29
0.17
0.81
1.99
-1.02
-0.84
-0.56
-0.29
0.17
1.06
1.99
-1.02
-0.84
-0.56
-0.20
0.17
1.08
1.99
-0.93
-0.75
-0.47
-0.20
0.17
1.45
2.27
-0.93
-0.75
-0.47
-0.20
0.35
1.45
2.27
Slide 30
Chebyshev’s Theorem
At least (1 - 1/k2) of the items in any data set will be
within k standard deviations of the mean, where k is
any value greater than 1.
• At least 75% of the items must be within
k = 2 standard deviations of the mean.
• At least 89% of the items must be within
k = 3 standard deviations of the mean.
• At least 94% of the items must be within
k = 4 standard deviations of the mean.
Slide 31
Example: Apartment Rents

Chebyshev’s Theorem
Let k = 1.5 with
x = 490.80 and s = 54.74
At least (1 - 1/(1.5)2) = 1 - 0.44 = 0.56 or 56%
of the rent values must be between
x - k(s) = 490.80 - 1.5(54.74) = 409
and
x + k(s) = 490.80 + 1.5(54.74) = 573
Slide 32
Example: Apartment Rents

Chebyshev’s Theorem (continued)
Actually, 86% of the rent values
are between 409 and 573.
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 33
Empirical Rule
For data having a bell-shaped distribution:
• Approximately 68% of the data values will be
within one standard deviation of the mean.
Slide 34
Empirical Rule
For data having a bell-shaped distribution:
• Approximately 95% of the data values will be
within two standard deviations of the mean.
Slide 35
Empirical Rule
For data having a bell-shaped distribution:
• Almost all (99.7%) of the items will be
within three standard deviations of the mean.
Slide 36
Example: Apartment Rents

Empirical Rule
Within +/- 1s
Within +/- 2s
Within +/- 3s
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
Interval
436.06 to 545.54
381.32 to 600.28
326.58 to 655.02
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
% in Interval
48/70 = 69%
68/70 = 97%
70/70 = 100%
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 37
Detecting Outliers





An outlier is an unusually small or unusually large
value in a data set.
A data value with a z-score less than -3 or greater
than +3 might be considered an outlier.
It might be an incorrectly recorded data value.
It might be a data value that was incorrectly included
in the data set.
It might be a correctly recorded data value that
belongs in the data set !
Slide 38
Example: Apartment Rents

Detecting Outliers
The most extreme z-scores are -1.20 and 2.27.
Using |z| > 3 as the criterion for an outlier,
there are no outliers in this data set.
Standardized Values for Apartment Rents
-1.20
-0.93
-0.75
-0.47
-0.20
0.35
1.54
-1.11
-0.93
-0.75
-0.38
-0.11
0.44
1.54
-1.11
-0.93
-0.75
-0.38
-0.01
0.62
1.63
-1.02
-0.84
-0.75
-0.34
-0.01
0.62
1.81
-1.02
-0.84
-0.75
-0.29
-0.01
0.62
1.99
-1.02
-0.84
-0.56
-0.29
0.17
0.81
1.99
-1.02
-0.84
-0.56
-0.29
0.17
1.06
1.99
-1.02
-0.84
-0.56
-0.20
0.17
1.08
1.99
-0.93
-0.75
-0.47
-0.20
0.17
1.45
2.27
-0.93
-0.75
-0.47
-0.20
0.35
1.45
2.27
Slide 39
Exploratory Data Analysis


Five-Number Summary
Box Plot
Slide 40
Five-Number Summary





Smallest Value
First Quartile
Median
Third Quartile
Largest Value
Slide 41
Example: Apartment Rents

Five-Number Summary
Lowest Value = 425
First Quartile = 450
Median = 475
Third Quartile = 525
Largest Value = 615
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 42
Box Plot



A box is drawn with its ends located at the first and
third quartiles.
A vertical line is drawn in the box at the location of
the median.
Limits are located (not drawn) using the interquartile
range (IQR).
• The lower limit is located 1.5(IQR) below Q1.
• The upper limit is located 1.5(IQR) above Q3.
• Data outside these limits are considered outliers.
… continued
Slide 43
Box Plot (Continued)


Whiskers (dashed lines) are drawn from the ends of
the box to the smallest and largest data values inside
the limits.
The locations of each outlier is shown with the
symbol * .
Slide 44
Example: Apartment Rents

Box Plot
Lower Limit: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5
Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5
There are no outliers.
37
5
40
0
42
5
45
0
47
5
50
0
52
5
550
575 600
625
Slide 45
Measures of Association
Between Two Variables


Covariance
Correlation Coefficient
Slide 46
Covariance



The covariance is a measure of the linear association
between two variables.
Positive values indicate a positive relationship.
Negative values indicate a negative relationship.
Slide 47
Covariance

If the data sets are samples, the covariance is denoted
by sxy.
 ( xi  x )( yi  y )
sxy 
n 1

If the data sets are populations, the covariance is
denoted
.
 xy by
 xy
 ( xi   x )( yi   y )

N
Slide 48
Correlation Coefficient




The coefficient can take on values between -1 and +1.
Values near -1 indicate a strong negative linear
relationship.
Values near +1 indicate a strong positive linear
relationship.
If the data sets are samples, the coefficient is rxy.
rxy 

sxy
sx s y
 xy
 xy 
If the data sets are populations,
the coefficient is  xy .
 x y
Slide 49
The Weighted Mean and
Working with Grouped Data




Weighted Mean
Mean for Grouped Data
Variance for Grouped Data
Standard Deviation for Grouped Data
Slide 50
Weighted Mean



When the mean is computed by giving each data
value a weight that reflects its importance, it is
referred to as a weighted mean.
In the computation of a grade point average (GPA),
the weights are the number of credit hours earned for
each grade.
When data values vary in importance, the analyst
must choose the weight that best reflects the
importance of each value.
Slide 51
Weighted Mean
x =  wi xi
 wi
where:
xi = value of observation i
wi = weight for observation i
Slide 52
Grouped Data




The weighted mean computation can be used to
obtain approximations of the mean, variance, and
standard deviation for the grouped data.
To compute the weighted mean, we treat the
midpoint of each class as though it were the mean of
all items in the class.
We compute a weighted mean of the class midpoints
using the class frequencies as weights.
Similarly, in computing the variance and standard
deviation, the class frequencies are used as weights.
Slide 53
Mean for Grouped Data

Sample Data
fM

x
f
i
fM


i
i
i

Population Data
i
N
where:
fi = frequency of class i
Mi = midpoint of class i
Slide 54
Example: Apartment Rents
Given below is the previous sample of monthly rents
for one-bedroom apartments presented here as grouped
data in the form of a frequency distribution.
Rent ($) Frequency
420-439
8
440-459
17
460-479
12
480-499
8
500-519
7
520-539
4
540-559
2
560-579
4
580-599
2
600-619
6
Slide 55
Example: Apartment Rents

Mean for Grouped Data
Rent ($)
420-439
440-459
460-479
480-499
500-519
520-539
540-559
560-579
580-599
600-619
Total
fi
8
17
12
8
7
4
2
4
2
6
70
Mi
429.5
449.5
469.5
489.5
509.5
529.5
549.5
569.5
589.5
609.5
f iMi
3436.0
7641.5
5634.0
3916.0
3566.5
2118.0
1099.0
2278.0
1179.0
3657.0
34525.0
34, 525
 493. 21
70
This approximation
differs by $2.41 from
the actual sample
mean of $490.80.
x
Slide 56
Variance for Grouped Data

Sample Data
2
f
(
M

x
)

i
i
s2 
n 1

Population Data
2
f
(
M


)

i
i
2 
N
Slide 57
Example: Apartment Rents

Variance for Grouped Data
s2  3, 017.89

Standard Deviation for Grouped Data
s  3, 017.89  54. 94
This approximation differs by only $.20
from the actual standard deviation of $54.74.
Slide 58
End of Chapter 3
Slide 59